# Superposition of two wave functions of different Hilbert spaces

I am trying to think of this problem for quite some time. Let's say, we have two sets of wave functions $\lbrace|\psi\rangle\rbrace$ and $\lbrace|\phi \rangle\rbrace$ and they belong to two different Hilbert spaces. That is,

$$\hat{H_1}|\psi\rangle=E_1|\psi\rangle$$

and

$$\hat{H_2}|\phi\rangle=E_2|\phi\rangle.$$

In the real space $\bf{R}$, their functional domains are disjoint. That is, if $\psi(x)$ is defined in $x\le0$, $\phi(x)$ is defined in $x>0$.

In this case, is it possible to conceive some kind of superposition between the two waves? If so, how? I mean how do we define the superposed wave function and what can be said about the energy? This paper introduces such a concept

http://dx.doi.org/10.1119/1.18854

• In quantum field theory, there is a field in spacetime. But in nonrelativistic quantum mechanics, a wavefunction is not a function from $\mathbb R^3$ into the complex numbers, a wave function is a function from the configuration space of the system into the joint spin state of the system. – Timaeus Sep 26 '15 at 17:34
• Where in the cited paper does it mention having two distinct Hilbert Spaces? It's a method of images solution to reflection off an infinite potential barrier. – Timaeus Sep 27 '15 at 1:25
• @Timaeus Isn't the wave function in quantum mechanics defined as $f_{\psi}\colon x\to \langle x|\psi\rangle$, and thus exactly a function from $\mathbb{R}$ to the complex numbers? – gented Sep 27 '15 at 18:55
• @GennaroTedesco Most definitely not. It is a function from configuration space into a joint spin state. The joint spin state is a tensor product of single particle spin states (one for each particle) and configuration space is $\mathbb R^{3n}$ when there are $n$ particles. – Timaeus Sep 27 '15 at 21:06
• How would you then define the wave function for the free spinless particle, according to that definition? Can you address me to some literature to check that? I have somehow never encountered such definition before. – gented Sep 27 '15 at 21:12

Two different Hibert spaces correspond to two different physical systems. Superposition of wave functions makes sence for one system (for one Hilbert space), since addition of vectors (quantum states) is defined in a particular vector space (Hilbert space). What you can do is to create a new Hilbert space by forming the tensor product of the two Hilbert spaces. And if you work in the coordinates representation, then you should expand the domain of the each wave function to the entire real line, by multiplying $\psi \left( x \right)$ whith the characteristic function of $\left( -\infty ,0 \right]$ and $\varphi \left( x \right)$ with the characteristic function of $\left( 0,+\infty \right)$.